Rössler with an Ornstein-Uhlenbeck control parameter
The Rössler flow driven through its bifurcations by a slow Ornstein-Uhlenbeck control parameter.
- Name:
ou_driven_rossler - Field:
mathematics/dynamical_systems - Formalism: stochastic differential equations
- References: Cestnik and Martens 2025; Rössler 1976; Uhlenbeck and Ornstein 1930
Factory
@register_model(
field='mathematics/dynamical_systems',
tags=('Rössler', 'Ornstein-Uhlenbeck', 'sde', 'bifurcation diagram', 'chaos', 'nonstationary'),
references=('Cestnik and Martens 2025', 'Rössler 1976', 'Uhlenbeck and Ornstein 1930'),
defaults={'until': 500.0, 'dt': 0.1, 'solver': EulerMaruyamaSolver(dt=0.01)},
)
def ou_driven_rossler(
x: float = 1.0,
y: float = 1.0,
z: float = 1.0,
*,
a: float = 0.2,
b: float = 0.4,
c: float = 5.743,
tau: float = 50.0,
rho: float = 0.5,
eta: float = 0.0,
) -> System:
"""Return the Rössler flow driven through its bifurcations by a slow Ornstein-Uhlenbeck control parameter.
The flow is `x' = -y - z`, `y' = x + a y + eta`, `z' = b + z (x - c)`, Eq. 13 of Cestnik and Martens, and the
control follows `eta' = (-eta + rho xi) / tau` with white noise `xi`. The control is an Ornstein-Uhlenbeck
process with the correlation time `tau` and the stationary variance `rho^2 / (2 tau)`. A control that varies
slowly against the period of the flow sweeps it through the period doublings and chaotic bands of its
bifurcation diagram.
Args:
x:
The initial first coordinate.
y:
The initial second coordinate.
z:
The initial third coordinate.
a:
The feedback of the second coordinate on itself.
b:
The constant drive of the third coordinate.
c:
The threshold of the third coordinate's growth.
tau:
The correlation time of the control.
rho:
The amplitude of the noise of the control.
eta:
The initial control.
Returns:
system:
The model, with the states `rossler.x`, `rossler.y`, `rossler.z`, and the control `rossler.eta`.
Examples:
>>> from skcomplex.models.mathematics.dynamical_systems import ou_driven_rossler
>>> ou_driven_rossler(tau=10.0).elements[0].tau
10.0
"""
rossler = DrivenRosslerElement('rossler', x=x, y=y, z=z, eta=eta, a=a, b=b, c=c, tau=tau, rho=rho)
drive = RateInteraction('drive', target='rossler', state='y', rate=lambda s: s.target.eta)
control = RelaxationInteraction(
'control',
target='rossler',
state='eta',
time=lambda s: s.target.tau,
noise=lambda s: s.target.rho / s.target.tau,
)
return System(elements=[rossler], interactions=[drive, control])
Components
class DrivenRosslerElement(RosslerElement):
"""Rössler flow with a control `eta`, the state that the drive adds to the rate of `y` and that relaxes to 0.
The element holds the flow and the states and parameters of its control. The interactions of the model add the
control to the rate of `y` and make it an Ornstein-Uhlenbeck process of correlation time `tau` and noise `rho`.
Args:
label:
The label of the element.
visual:
How the element is drawn, or `None` for `default_visual_`.
x:
The initial first coordinate.
y:
The initial second coordinate.
z:
The initial third coordinate.
eta:
The initial control.
a:
The feedback of the second coordinate on itself.
b:
The constant drive of the third coordinate.
c:
The threshold of the third coordinate's growth.
tau:
The correlation time of the control.
rho:
The amplitude of the noise of the control.
Examples:
>>> from skcomplex.models.mathematics.dynamical_systems import DrivenRosslerElement
>>> DrivenRosslerElement('rossler', tau=10.0).tau
10.0
"""
def __init__(
self,
label: str,
visual: VisualSpec | None = None,
*,
x: float = 1.0,
y: float = 1.0,
z: float = 1.0,
eta: Annotated[float, StateSpec()] = 0.0,
a: float = 0.2,
b: float = 0.4,
c: float = 5.743,
tau: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 50.0,
rho: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.5,
) -> None:
super().__init__(label, visual=visual, x=x, y=y, z=z, a=a, b=b, c=c)
self.eta = eta
self.tau = tau
self.rho = rho
class RateInteraction(Interaction):
"""Rate `rate(ns)` and noise `noise(ns) dW` added to the highest time derivative of a state of a target.
On a state of order 1 the rate is its first derivative, and on a state of order `k` it adds to the `k`-th
derivative, so rates on one state superpose. With a divisor `d` the rate and the noise are divided by `d`, the name
of a state or parameter of the target, such as an inertia, a friction, or a capacitance, which backends can read.
A rate, a noise, or a divisor function takes the namespace of the interaction, with the namespaces `target` and
`sources` of the bound elements, the time `t`, and the symbols of the parameters a subclass declares. A number
stands for a constant, and a number on a vector state applies to every component. The noise is a Wiener process per
component and member by default, a process of a `WienerProcessSpec` to share or correlate it across states and
members, or a fractional Brownian motion of a Hurst index.
Args:
label:
The label of the interaction, unique among the parts of its system.
target:
The path of the element or group whose state the rate changes.
state:
The name of the state of the target that the rate changes.
rate:
A function of the namespace that returns the rate, or a number.
sources:
The paths of the elements or groups the rate and the noise read but do not change.
noise:
A function of the namespace that returns the amplitude of the noise, a number, or `None` for no noise.
divisor:
The name of the state or parameter of the target that divides the rate and the noise, a function of the
namespace of the target that returns the divisor, such as a moment of inertia `m l^2`, or `None`.
process:
The Wiener processes that drive the noise, or `None` for one independent process per component and member.
index:
The process of `process` that drives the noise, or `None` for every process, one per component of a vector
state.
convention:
The stochastic calculus of the noise, `'ito'`, `'stratonovich'`, or `'anti-ito'`.
hurst:
The Hurst index in `(0, 1)` of a fractional Brownian motion that drives the noise, a number or a function
of the namespace that returns a parameter, or `None` for a Wiener process.
conserved:
Whether the noise is the divergence of a random flux on the links of the lattice of a field, so it keeps the
total of the state.
Raises:
ValidationError:
The state is not a name, a function is neither callable nor a number, the convention is unknown, the Hurst
index is outside `(0, 1)`, or the noise options cannot combine.
Examples:
>>> from skcomplex.core.elements import CompartmentElement
>>> from skcomplex.core.interactions import RateInteraction
>>> from skcomplex.core.system import System
>>> from skcomplex.engine.runtime import Simulation
>>> decay = RateInteraction('decay', target='stock', state='n', rate=lambda s: -0.5 * s.target.n)
>>> system = System(elements=[CompartmentElement('stock', n=8.0)], interactions=[decay])
>>> trajectory = Simulation(system, dt=0.01).check().prepare().run(until=2.0).trajectory_
>>> round(float(trajectory.values('stock.n')[-1]), 3)
2.943
"""
def __init__(
self,
label: str,
*,
target: Annotated[str, RoleSpec()],
state: str = 'x',
rate: Callable[[SimpleNamespace], object] | float = 0.0,
sources: Annotated[Sequence[str], RoleSpec(variadic=True)] = (),
noise: Callable[[SimpleNamespace], object] | float | None = None,
divisor: str | Callable[[SimpleNamespace], object] | None = None,
process: Annotated[WienerProcessSpec | None, RoleSpec(noise=True)] = None,
index: int | None = None,
convention: str = 'ito',
hurst: Callable[[SimpleNamespace], object] | float | None = None,
conserved: bool = False,
) -> None:
super().__init__(label)
self.target = target
self.state = state
self.rate = rate
self.sources = sources
self.noise = noise
self.divisor = divisor
self.process = process
self.index = index
self.convention = convention
self.hurst = hurst
self.conserved = conserved
def _check_functions(self) -> None:
"""Raise an error for a state that is not a name, or a function that is neither callable nor a number."""
if not isinstance(self.state, str) or not self.state.isidentifier():
error_msg = (
f'{type(self).__name__} {self.label!r} changes the state {self.state!r}, which is not a name of a '
'state.'
)
raise ValidationError(error_msg)
for name in ('rate', 'noise'):
value = getattr(self, name)
if not (value is None or callable(value) or isinstance(value, int | float)):
error_msg = f'The {name} of {self.label!r} is {value!r}. Give a function of the namespace or a number.'
raise ValidationError(error_msg)
if not (
self.divisor is None
or callable(self.divisor)
or (isinstance(self.divisor, str) and self.divisor.isidentifier())
):
error_msg = (
f'The divisor of {self.label!r} is {self.divisor!r}. Give the name of a state or parameter, or a '
'function of the namespace of the target.'
)
raise ValidationError(error_msg)
def _check_noise(self) -> None:
"""Raise an error for an unknown convention, a Hurst index outside `(0, 1)`, or noise options that conflict."""
if self.convention not in CONVENTIONS:
error_msg = f'Unknown noise convention {self.convention!r}. Use one of {", ".join(CONVENTIONS)}.'
raise ValidationError(error_msg)
if not (
self.hurst is None or callable(self.hurst) or (isinstance(self.hurst, int | float) and 0 < self.hurst < 1)
):
error_msg = f'The Hurst index of {self.label!r} is {self.hurst!r}. Give a number in (0, 1) or a function.'
raise ValidationError(error_msg)
if self.index is not None and self.process is None:
error_msg = f'{self.label!r} picks the process {self.index!r} but binds no processes. Give `process`.'
raise ValidationError(error_msg)
options = [
name
for name, given in (('hurst', self.hurst), ('process', self.process), ('conserved', self.conserved))
if given
]
if len(options) > 1:
error_msg = f'The noise of {self.label!r} takes {" and ".join(options)}. Give at most one of them.'
raise ValidationError(error_msg)
def _check(self) -> None:
"""Raise an error for a state, functions, or noise options the interaction cannot take.
Raises:
ValidationError:
The type declares states, the state is not a name, a function is neither callable nor a number, the
convention is unknown, the Hurst index is outside `(0, 1)`, or the noise options cannot combine.
"""
super()._check()
self._check_functions()
self._check_noise()
def _evaluate(self, function: Callable[[SimpleNamespace], object] | float, namespace: SimpleNamespace) -> object:
"""Return the value of a function of the namespace, or a number as it is."""
return function(namespace) if callable(function) else function
def _broadcast(self, value: object, state: object) -> object:
"""Return a value with one component per component of a vector state, repeating a number."""
if isinstance(state, FreeModuleElement) and not isinstance(value, FreeModuleElement | list | tuple):
return vector(SR, [value] * len(state))
if isinstance(value, list | tuple):
return vector(SR, value)
return value
def _divide(self, value: object, namespace: SimpleNamespace) -> object:
"""Return a value divided by the divisor of the target, or the value without a divisor."""
if self.divisor is None:
return value
divisor = (
self.divisor(self._get_member(namespace, 'target'))
if callable(self.divisor)
else getattr(self._get_member(namespace, 'target'), self.divisor)
)
return operator.truediv(value, divisor)
def _get_rate(self, namespace: SimpleNamespace) -> object:
"""Return the rate before the divisor, the hook a subclass overrides to derive it from its parameters."""
return self._evaluate(self.rate, namespace)
def _get_noise(self, namespace: SimpleNamespace) -> object:
"""Return the noise amplitude before the divisor, the hook a subclass overrides, or `None` for no noise."""
return None if self.noise is None else self._evaluate(self.noise, namespace)
def _get_noise_law(self, state: object, amplitude: object, namespace: SimpleNamespace) -> BasePart:
"""Return the noise law of an amplitude, by the process, the convention, and the Hurst index."""
if self.hurst is not None:
hurst = self._evaluate(self.hurst, namespace)
return FractionalNoiseLaw(state, amplitude, hurst=hurst, convention=self.convention)
process = None
if self.process is not None:
handle = self._get_member(namespace, 'process').dW
process = handle if self.index is None else handle[self.index]
return NoiseLaw(state, amplitude, convention=self.convention, process=process, conserved=self.conserved)
def _get_laws(self, namespace: SimpleNamespace) -> list[BasePart]:
"""Return the rate law of the state and its noise law, both divided by the divisor.
Args:
namespace:
The namespace of the symbols the laws can use, with the namespaces of the target and the sources.
Returns:
laws:
The rate law, and the noise law when the interaction has noise.
"""
state = getattr(self._get_member(namespace, 'target'), self.state)
rate = self._divide(self._broadcast(self._get_rate(namespace), state), namespace)
laws: list[BasePart] = [RateLaw(state, rate)]
noise = self._get_noise(namespace)
if noise is not None:
laws.append(self._get_noise_law(state, self._divide(noise, namespace), namespace))
return laws
class RelaxationInteraction(RateInteraction):
"""Relaxation `(goal(ns) - x) / time(ns)` of a state `x` of a target towards a goal, with optional noise.
With a constant goal and time and an additive noise of amplitude `sigma`, the state follows the Ornstein-Uhlenbeck
process `dx = (goal - x) / time dt + sigma dW`, whose stationary variance is `sigma**2 time / 2`. Without noise it
is first-order smoothing, or adaptive expectation, towards the goal. A goal or a time function takes the namespace
of the interaction, as for `RateInteraction`, and a number stands for a constant. A goal that is a number applies
to every component of a vector state.
Args:
label:
The label of the interaction, unique among the parts of its system.
target:
The path of the element or group whose state relaxes.
state:
The name of the state of the target that relaxes.
goal:
A function of the namespace that returns the goal, or a number.
time:
A function of the namespace that returns the relaxation time, or a positive number.
noise:
A function of the namespace that returns the amplitude of the noise, a number, or `None` for no noise.
Raises:
ValidationError:
The state is not a name, or a function is neither callable nor a number.
Examples:
>>> from skcomplex.core.elements import CompartmentElement
>>> from skcomplex.core.interactions import RelaxationInteraction
>>> from skcomplex.core.system import System
>>> from skcomplex.engine.runtime import Simulation
>>> smoothing = RelaxationInteraction('smoothing', target='belief', state='n', goal=10.0, time=0.5)
>>> system = System(elements=[CompartmentElement('belief', n=0.0)], interactions=[smoothing])
>>> trajectory = Simulation(system, dt=0.01).check().prepare().run(until=5.0).trajectory_
>>> round(float(trajectory.values('belief.n')[-1]), 3)
10.0
"""
def __init__(
self,
label: str,
*,
target: Annotated[str, RoleSpec()],
state: str = 'x',
goal: Callable[[SimpleNamespace], object] | float = 0.0,
time: Callable[[SimpleNamespace], object] | float = 1.0,
noise: Callable[[SimpleNamespace], object] | float | None = None,
) -> None:
super().__init__(label, target=target, state=state, noise=noise)
self.goal = goal
self.time = time
def _check_functions(self) -> None:
"""Raise an error for a state that is not a name, or a function that is neither callable nor a number."""
super()._check_functions()
for name in ('goal', 'time'):
value = getattr(self, name)
if not (callable(value) or isinstance(value, int | float)):
error_msg = f'The {name} of {self.label!r} is {value!r}. Give a function of the namespace or a number.'
raise ValidationError(error_msg)
def _get_goal(self, namespace: SimpleNamespace) -> object:
"""Return the goal, the hook a subclass overrides to derive it from its parameters."""
return self._evaluate(self.goal, namespace)
def _get_time(self, namespace: SimpleNamespace) -> object:
"""Return the relaxation time, the hook a subclass overrides to derive it from its parameters."""
return self._evaluate(self.time, namespace)
def _get_rate(self, namespace: SimpleNamespace) -> object:
"""Return the relaxation `(goal - x) / time` of the state.
Args:
namespace:
The namespace of the symbols the rate can use.
Returns:
rate:
The rate of the state.
"""
state = getattr(self._get_member(namespace, 'target'), self.state)
goal = self._broadcast(self._get_goal(namespace), state)
return (goal - state) / self._get_time(namespace)
Equations
rossler.x: diff(x_rossler(t), t) == -y_rossler(t) - z_rossler(t)
rossler.y: diff(y_rossler(t), t) == a_rossler*y_rossler(t) + eta_rossler(t) + x_rossler(t)
rossler.z: diff(z_rossler(t), t) == -(c_rossler - x_rossler(t))*z_rossler(t) + b_rossler
rossler.eta: diff(eta_rossler(t), t) == -eta_rossler(t)/tau_rossler